Complex Numbers (Book)

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Complex Numbers

Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

http://www.ddewey.net/mandelbrot/

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Show Recent Comments

10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 8 hours ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

a day ago
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

a day ago
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

a day ago
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

September 4, 2024 at 10:43am

#1076238

Isn't It Draconic?

"Blog Week Birthday Bastion 2024"

[13+] by WakeUpAndLive

Prompt 4. Sept 4.

Photo prompts. (Choose 1,2,3 or 4 and write fiction or non-fiction)

PhotoPrompt 3

Photo Prompt 3

I'm only posting the prompt pic I picked. One of the others was a blue rubber duck, and I almost picked that one, but dismissed it as "too cute."

Ducks and dragons are related, though.

A drake is a male duck. A drake is also, depending on the source, a particular subtype of dragon, or a synonym for dragon. The latter has a different etymology, one which is more apparent when you consider the adjective form, "draconic."

That should not be confused with "draconian," which is used to describe oppressive policies. Nor should it be confused with the explorer Sir Francis Drake, or his namesake Frank Drake, originator of the Drake Equation, which has nothing to do with dragons. Unless aliens are dragons. Which isn't technically impossible.

Another synonym for dragon is worm, or wyrm, which also confused me as a kid. English is weird.

Dragons are, unlike ducks, creatures of fantasy. Unless you count Komodo dragons, though I say that's cheating. But I have this hypothesis that dragons were dinosaurs.

Why were dragons part of the folklore of several geographically separate cultures? I think it's because of dinosaur bones. Some ancient t-rex or brontosaurus skeleton or whatever got exposed via erosion. Early humans, clever and creative but without our understanding of geological time, might have seen these enormous bones poking out of some sedimentary rock, and pictured what sort of creature might have made them. Clearly, it was a giant, flying, fire-breathing, gold-hoarding monster, because what else could it be?

But also, ducks are dinosaurs.

All birds are. Lots of dinosaur lineages indeed went extinct, but at least one survived into modern times and their genetically-plastic descendants exploded into the vast diversity of avian species we encounter today. Those eggs you had for breakfast? Dinosaur eggs. That KFC bucket you reluctantly bought over the Labor Day weekend? Kentucky-fried dinosaur parts.

Don't feel bad about eating chicken. Their ancestors probably munched on ours. Colonel Sanders was just doling out payback.

So, if you're wondering what a dragon would taste like... well, it would probably taste like chicken.

Or duck.

Unlike our modern birds, though, dragon meat might even come pre-cooked. All that fire breath had to do something to their insides, right?

Let's do this again. We get to give out a free Merit Badge every day this week. Want one? Anyone who comments here before 11:30 pm WDC time today could get today's. (I'll need that extra half-hour to pick a winner and send the badge before midnight.)

To clarify:

When I say "comment," I mean comment. Not review. Though reviews are always welcome.

I also mean "here," not on the newsfeed post.

MB recipient will be chosen at random.

Maximum of one MB per commenter for the week.

If I don't get comments, I'll pick a previous commenter, and maybe not at random.

The MB will be the one I commissioned two years ago, "Complexity," which is a publicly available MB.

I appreciate all comments; this is just a little incentive.

5 Like • Comment • View 7 comments

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